What is "measurable function"

n. 1 (context analysis English) If a function's codomain is a topological space and the function's domain is a measurable space, then the function is measurable if the inverse image of every open set in its codomain is a measurable set in its domain. 2 (context mathematics English) Any well-behaved function of real numbers between measurable spaces.

In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration. Specifically, a function between measurable spaces is said to be measurable if the preimage of each measurable set is measurable, analogous to the situation of continuous functions between topological spaces. A measurable function is said to be bimeasurable if it is bijective and its inverse is also measurable.

In probability theory, the σ-algebra often represents the set of available information, and a function (in this context a random variable) is measurable if and only if it represents an outcome that is knowable based on the available information. In contrast, functions that are not Lebesgue measurable are generally considered pathological, at least in the field of analysis.